On a non-linear droplet oscillation theory via the unified method

Abstract

Presently, the oscillation of a liquid droplet in a dynamically negligible outer medium subject to surface tension and small viscosity is investigated. By using the potential flow assumption, the unified transform method by Fokas is employed to reduce the corresponding free boundary problem formulated on a time-dependent domain into a nonlinear system of integro-differential equations (IDEs). This new system depends on one less spatial variable and is now defined on a time-independent domain. Most importantly, the resulting set of equations governs the general droplet oscillation with arbitrarily large deviations from the spherical shape. As the nonlinearity of the above IDE system up to now prevented an analytical solution, the Poincaré expansion technique is employed, retaining terms up to the second order. By decomposing the unknowns into normal modes, these equations are uncoupled and the resulting ordinary differential equations for the mode amplitudes are solved, and the results are compared to those of previous works. It should be stressed that the present analysis is limited to small viscosity, or, in other words, for small Ohnesorge numbers. The reason for this is that, inside of the droplet, a potential flow is assumed and the viscous effect is taken into account only at the droplet surface by the jump condition of momentum. This is only reasonable for a small viscosity and a short time. Otherwise, vorticity is generated at the interface and diffuses toward the inside of the droplet.

Application of discrete mechanics model to jump conditions in two-phase flows

“…More recent studies, including [20], [44], [45], [46], have extended the field of investigation to dissipative effects using linear theories. The problem of the oscillation of a droplet in nonlinear theory for a small Ohnesorge number has been addressed recently [47] using the potential flow assumption to reduce the corresponding free boundary problem formulated on a time-dependent domain into a nonlinear system of integro-differential equations. For two-dimensional planar drops oscillating about a circle, the frequency of the oscillations is given by:…”

The Modified Helmholtz Equation on a Regular Hexagon—The Symmetric Dirichlet Problem

“…The global relation has had important analytical and numerical implications: first, it has led to novel analytical formulations of a variety of important physical problems from water waves [20][21][22][23][24][25][26] to three-dimensional layer scattering [27]. Second, it has led to the development of new techniques for the Laplace, modified Helmholtz, Helmholtz, biharmonic equations, both analytical [28][29][30][31][32][33][34][35] and numerical [36][37][38][39][40][41][42][43][44][45][46][47].…”